Practice (90/1000)
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A square is inscribed in a circle of radius 5 units. In each of the four regions bounded by a side of the square and the smaller circular arc joining the endpoints of that side, a square is drawn so that one side lies on the side of the larger square and the two opposite vertices lie on the circle, as shown. What is the total area of the five squares? Express your answer to the nearest whole number.
A square and a regular hexagon are coplanar and share a common side as shown. What is the sum of the degree measures of angles 1 and 2?
$\triangle{ABC}$ has vertices at A(-3, 4), B(5, 0) and C(1, -4). What is the $x$-coordinate of the point where the median from C intersects $\overline{AB}$?
The sum of three primes is 125. The difference between the largest and the smallest is 50. What is the largest possible median of these three prime numbers?
What is the probability that a randomly selected integer from $1$ to $81$, inclusive, is equal to the product of two one-digit numbers?
Shaina has one stick of length $a$ cm and another of length $b$ cm, where $a \ne b$. She needs a third stick with length strictly between 8 cm and 26 cm to make the third side of a triangle. What is the product $ab$?
A 3-inch by 8-inch sheet of paper and a 2-inch by 12-inch sheet of paper have the same area. Using just one cut (not necessarily straight), the 3-inch by 8-inch sheet can be divided into two pieces that can be rearranged to completely cover the 2-inch by 12-inch sheet. What is the length of the cut?
How many diagonals does a convex octagon have?
Jean is twice as likely to make a free throw as she is to miss it. What is the probability that she will miss $3$ times in a row?
A pyramid has 6 vertices and 6 faces. How many edges does it have?
The product of the integers from 1 through 7 is equal to $2^j\cdot 3^k\cdot 5 \cdot 7$ What is the value of $j - k$?
Let's name the coordinates of the vertices of a trapezoid are A(1, 7), B(1, 11), C(8, 4) and D(4, 4). What is the area of the trapezoid?
Four consecutive integers are substituted in every possible way for distinct values $a$, $b$, $c$ and $d$. What is the positive difference between the smallest and largest possible values of $(ab + cd)$?
Triangle $\triangle{MNO}$ is an isosceles trianglewith MN = NO = 25. A line segment drawn from the midpoint of MO perpendicular to MN, intersects MN at point P with NP:PM = 4:1. We must find the length of the altitude drawn from point N to side MO.
In a sequence of positive integers, every term after the first two terms is the sum of the previous two terms of the sequence. The fifth term is 2012 so what is the maximum possible value of the first term?
The figure shows the first three stages of a fractal, respectively. We must find how many circles in Stage 5 of the fractal.
We have a set of numbers {1, 2, 3, 4, 5} and we take products of three different numbers. We must find now many pairs of relatively prime numbers there are.
In rectangle $ABCD$, $AB = 6$ units. m$\angle{DBC} = 30^{\circ}$, $M$ is the midpoint of segment $AD$, and segments $CM$ and $BD$ intersect at point $K$. We must find the length of segment $MK$.
What is the largest five-digit integer such that the product of the digits is $2520$?
A rectangular prism is composed of unit cubes. The outside faces of the prism are painted blue and the seven unit cubes in the interior are unpainted. We must find how many unit cubes have exactly one painted face.
Let $f(x) = x^2 + 5$, and $g(x) = 2(f(x))$. What is the greatest possible value of $f(x + 1)$ when $g(x)$ = 108?
A right triangle has sides with lengths 8, 15 and 17. A circle is inscribed in the triangle, as shown, and we must find the radius of the circle.
In trapezoid ABCD segments AB and CD are parallel. Point P is the intersection of diagonals AC and BD. The area of $\triangle{PAB}$ is 16 and $\triangle{PCD}$ is 25. We must find the area of the trapezoid.
The sum of the squares of two positive numbers is 20 and the sum of their reciprocals is 2. We must find their product.
A triangle has angles measuring $15^{\circ}$, $45^{\circ}$ and $120^{\circ}$. The side opposite the $45^{\circ}$ angle is 20 units. The area of the triangle can be expressed as $m -n\sqrt{q}$ and we must find the sum $m + n + q$.